The version of windows 7 I bought came with both CDs, one for 32 and one for 64 - so even if your pc needed 32, your win7 license could later be reinstalled as 64b on a new machine.

Not all packages in not all store come with both, though, so best to check

I found that getting network stuff like services access etc. was one area where 7 can be fiddly. Homegroup functionality for example can get in the way.

Things it could be:
- apache problems. Some file or certificate or some part of the config is unhappy with the machine change
- firewall/malware scanner problems
- tcpip configuration
- connectivity to mysql, either a config needs changing or the mysql user access settings
- interference with tools, sometimes these beautiful manufacturer support service tools are php/apache based. unlikely that they use the port you do, but worth a check
- i can probably keep thinking of more but hopefully won't need to

MS has a primitive network troubleshooter that should have a tool to check the setup for remote access
There should be a security event log in the event viewer (except I cannot see the network log here on mine) - should help you spot if it is one of the core windows features interfering or not. It might not be, so worth checking

"connection was reset" can mean either that something prevented you from connecting, or that your server request errored and died, so it might be worth checking the apache error log to eliminate that possibility. It might give you a good hint to look at the access log as well and see if your server receives the requests or not, that eliminates a lot of possibilities if we know which